"We are given:" We are given:
\qquad R \ = \ ZZ[ \sqrt{2} ]; \qquad \quad M \ = \ { a + b \sqrt{2} \in R \ | \quad 5 | a \quad "and" \quad 5 | b }.
"i) We want to show:" \qquad M \ \ "is an ideal of" \ \ R.
"We proceed as follows. We need to show:"
\qquad \qquad \qquad p, q \in M \quad "and" \quad r, s \in R \quad rArr \quad r p + s q \in M.
"Suppose:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad p, q \in M \quad "and" \quad r, s \in R .
"So, by definition:"
p = 5 a + 5 b \sqrt{2}, \ q = 5 c + 5 d \sqrt{2}; \ r = f + g \sqrt{2}, \ s = h + k \sqrt{2};
\qquad \qquad \qquad "where:" \qquad \qquad \qquad a, b, c, d, f, g, h, k \in ZZ.
"Now we compute:"
\ r p + s q \ = \ ( f + g \sqrt{2} ) ( 5 a + 5 b \sqrt{2} ) + ( h + k \sqrt{2} ) ( 5 c + 5 d \sqrt{2} )
\qquad = \ ( 5 f a + 5 f b \sqrt{2} + 5 g a \sqrt{2} + 5 g b ( \sqrt{2} )^2 )
\qquad \qquad \qquad + ( 5 h c + 5 h d \sqrt{2} + 5 k c \sqrt{2} + 5 k d ( \sqrt{2} )^2 )
\qquad = \ ( 5 f a + 5 f b \sqrt{2} + 5 g a \sqrt{2} + 10 g b )
\qquad \qquad \qquad + ( 5 h c + 5 h d \sqrt{2} + 5 k c \sqrt{2} + 10 k d )
\qquad = \ ( 5 f a + 5 h c + 10 g b + 10 k d )
\qquad \qquad \qquad + ( 5 f b \sqrt{2} + 5 h d \sqrt{2} + 5 g a \sqrt{2}+ 5 k c \sqrt{2} )
\qquad = \ 5 ( f a + h c + 2 g b + 2 k d )
\qquad \qquad \qquad + 5 ( f b + h d + g a + k c ) \sqrt{2}
\qquad = \ 5 A + 5 B \sqrt{2}; \qquad "and" \quad A, B \in ZZ.
"So:"
\qquad \qquad \qquad \quad \ r p + s q \ = \ 5 A + 5 B \sqrt{2}; \qquad "and" \quad A, B \in ZZ.
"Thus, by definition of" \ M ":"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad r p + s q \in M.
"Thus:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad M \ \ "is an ideal of" \ \ R. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad square
"ii) Here, we want to show:"
\qquad \qquad \qquad \qquad \qquad 5 cancel{|} a \qquad "or" \qquad 5 cancel{|} b \quad rArr \quad 5 cancel{|} ( a^2 + b^2 ).
"We proceed as follows. What need to show is the same as"
"showing the following:"
\qquad \qquad \qquad \qquad \qquad \qquad 5 | ( a^2 + b^2 ) \quad rArr \quad 5|a \quad "and" \quad 5|b.
"This statement is false !! Here is a counter-example:"
\quad \ \ 5 | 25 \quad rArr \quad 5|( 3^2 + 4^2 ); \qquad "but certainly" \quad \ 5 cancel{|} 3 \quad "and" \quad 5 cancel{|} 4.
